Abstract
We extend Atanassov’s methods for Halton sequences in two different directions: (1) in the direction of Niederreiter (t, s) − sequences, (2) in the direction of generating matrices for Halton sequences. It is quite remarkable that Atanassov’s method for classical Halton sequences applies almost “word for word” to (t, s) − sequences and gives an upper bound quite comparable to those of Sobol’, Faure, and Niederreiter. But Atanassov also found a way to improve further his bound for classical Halton sequences by means of a clever scrambling producing sequences which he named modified Halton sequences. We generalize his method to nonsingular lower triangular matrices in the last part of this article.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
E. I. Atanassov, On the discrepancy of the Halton sequences, Math. Balkanica, New Series 18.1–2 (2004), 15–32.
E. I. Atanassov and M. Durchova, Generating and testing the modified Halton sequences. In Fifth International Conference on Numerical Methods and Applications, Borovets 2002, Springer-Verlag (Berlin), Lecture Notes in Computer Science 2542 (2003), 91–98.
J. Dick and F. Pillichshammer, Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, UK (2010).
H. Faure, Discrépance de suites associées à un système de numération (en dimension un), Bull. Soc. math. France 109 (1981), 143–182.
H. Faure, Discrépance de suites associées à un système de numération (en dimension s), Acta Arith. 61 (1982), 337–351.
H. Faure, On the star-discrepancy of generalized Hammersley sequences in two dimensions, Monatsh. Math. 101 (1986), 291–300.
H. Faure, Méthodes quasi-Monte Carlo multidimensionnelles, Theoretical Computer Science 123 (1994), 131–137.
H. Faure and C. Lemieux, Generalized Halton sequences in 2008: A comparative study, ACM Trans. Model. Comp. Sim. 19 (2009), Article 15.
H. Faure and C. Lemieux, Improvements on the star discrepancy of (t, s) − sequences. Submitted for publication, 2011.
J. H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math. 2 (1960), 184–90.
R. Hofer and G. Larcher, On the existence and discrepancy of certain digital Niederreiter–Halton sequences, Acta Arith. 141 (2010), 369–394.
P. Kritzer, Improved upper bounds on the star discrepancy of (t, m, s)-nets and (t, s)-sequences, J. Complexity 22 (2006), 336–347.
C. Lemieux, Monte Carlo and Quasi-Monte Carlo Sampling, Springer Series in Statistics, Springer, New York (2009).
H. Niederreiter, Point sets and sequences with small discrepancy, Monatsh. Math. 104 (1987), 273–337.
H. Niederreiter and F. Özbudak, Low-discrepancy sequences using duality and global function fields, Acta Arith. 130 (2007), 79–97.
H. Niederreiter and C. P. Xing, Quasirandom points and global function fields, Finite Fields and Applications, S. Cohen and H. Niederreiter (Eds), London Math. Soc. Lectures Notes Series 233 (1996), 269–296.
S. Tezuka, Polynomial arithmetic analogue of Halton sequences, ACM Trans. Modeling and Computer Simulation 3 (1993), 99–107.
S. Tezuka, A generalization of Faure sequences and its efficient implementation, Technical Report RT0105, IBM Research, Tokyo Research Laboratory (1994).
X. Wang, C. Lemieux, H. Faure, A note on Atanassov’s discrepancy bound for the Halton sequence, Technical report, University of Waterloo, Canada (2008). Available at sas.uwaterloo.ca/stats_navigation/techreports/08techreports.shtml.
Acknowledgements
We wish to thank the referee for his/her detailed comments, which were very helpful to improve the presentation of this manuscript. The second author acknowledges the support of NSERC for this work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Faure, H., Lemieux, C., Wang, X. (2012). Extensions of Atanassov’s Methods for Halton Sequences. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-27440-4_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-27439-8
Online ISBN: 978-3-642-27440-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)